Bounds & Error Intervals, Knowledge Organiser

Key Terms

• Lower bound (LB): Smallest value that rounds to the given value
• Upper bound (UB): Smallest value that would round UP to the next value
• Error interval: Range of possible values written as an inequality
• Degree of accuracy: The precision of a measurement (e.g. nearest 10, 1 d.p.)
• Half interval: Half the degree of accuracy: added/subtracted to find bounds

Must-Know Facts

• Error interval: LB ≤ x < UB (lower uses ≤; upper uses <)
• LB = value − half interval; UB = value + half interval
• Rounded to nearest 10 → half interval = 5
• Rounded to 1 d.p. → half interval = 0.05
• Maximum of A + B: UB(A) + UB(B)
• Maximum of A ÷ B: UB(A) ÷ LB(B) (bigger top, smaller bottom)
• Minimum of A ÷ B: LB(A) ÷ UB(B)

Higher Level: Bounds in Calculations

• Max A + B: UB(A) + UB(B)
• Min A + B: LB(A) + LB(B)
• Max A − B: UB(A) − LB(B)
• Min A − B: LB(A) − UB(B)
• Max A × B: UB(A) × UB(B)
• Max A ÷ B: UB(A) ÷ LB(B)

Key Formulas

• Lower bound = rounded value − ½ × degree of accuracy
• Upper bound = rounded value + ½ × degree of accuracy
• Error interval: LB ≤ x < UB
• Max sum = UB(A) + UB(B); Min sum = LB(A) + LB(B)
• Max difference = UB(A) − LB(B); Min difference = LB(A) − UB(B)

Common Mistakes

• Error interval notation: Use ≤ x < UB (upper bound is strict inequality), not ≤ x ≤ UB
• Max difference: Subtract the LOWER bound of B from UPPER bound of A, not both upper bounds
• Half the accuracy: For a measurement to 1 decimal place, the accuracy is 0.1 so bounds are ±0.05, not ±0.1
• Max division: Divide by the LOWER bound of the denominator to maximise the result
• Significant figures vs decimal places: Check which is stated: they give different bound widths

Worked Examples

A length is 4.7 m correct to 1 decimal place. Write down the error interval. Step 1 Find half the accuracy 1 d.p. means accuracy of 0.1 m Half = 0.05 m Step 2 Calculate bounds LB = 4.7 − 0.05 = 4.65 UB = 4.7 + 0.05 = 4.75 Step 3 Write error interval 4.65 ≤ length < 4.75

Worked Examples

Distance = 100 km (nearest 10 km), Time = 2 hours (nearest hour). Find the minimum speed. Step 1 Find bounds for each Distance: LB = 95, UB = 105 Time: LB = 1.5, UB = 2.5 Step 2 For minimum speed = min distance ÷ max time Min speed = LB(distance) ÷ UB(time) = 95 ÷ 2.5 = 38 km/h